Optical system design method using real number surface number

ABSTRACT

Generally, it is difficult to previously know an adequate position of a special optical surface such as an aspherical surface, a diffraction optical element, or an eccentric surface in an optical system. If a trial of designing an optical system with all the possible surface numbers is made, the number of combinations is very large, and the designing is often impossible. The invention solves such a problem and provides means for automatically and efficiently finding out the most suitable surface number of a special optical surface. According to the invention, the surface number of a special optical surface different from a spherical surface is expanded to a real number value, the constitution of an optical system including a special optical surface having a real number value surface number is defined, the real number value surface number is used as an independent variable for optimizing the optical system, and the best surface number of the special optical surface is determined. If the real number value surface number lies in the range from an integer n to an integer n+1, one method for defining the constitution of an optical system including the special optical surface is inserting one or more virtual optical surfaces between an optical surface n and an optical surface n+1 and setting the virtual optical surfaces as special optical surfaces determined by the characteristic values and the real number value surface numbers of the special optical surfaces and different from a spherical surface.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of International Application No.PCT/JP2005/008781, with International Filing Date of May 13, 2005.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of optimizing an opticalsystem which realizes automated design of an optical system.

2. Description of the Related Art

Optical surfaces in an optical system are usually spheres which havetheir centers on the optical axis. Special surfaces such as aspheres,diffractive optical elements (DOEs), and decentered surfaces areeffective for improving the function and the quality of the opticalsystem and reducing the weight, size, and cost. But these specialsurfaces tend to be more expensive and more sensitive to manufacturingerrors than ordinary spheres. Then it is important to use the limitednumber of special surfaces at the appropriate positions in the opticalsystem.

But in general it is difficult to know the appropriate positions ofspecial surfaces in the optical system a priori. In the actual design,it is common to try a number of combinations of the positions of specialsurfaces and then to choose the best result from them. If all thepossible combinations are to be tried, the cases are usually too many tobe tried actually. Considering this problem a new method is proposedhere to find the best positions of special surfaces automatically andefficiently.

BRIEF SUMMARY OF THE INVENTION

This method extends the surface number of a special surface to a realnumber and defines the construction of the optical system with opticalsurfaces with real number surface numbers. The best surface numbers ofspecial surfaces are determined by including the real number surfacenumbers as independent variables of the optimization of the opticalsystem.

When the surface number of an optical surface with a real number surfacenumber is an integer n, a procedure is defined to set up the opticalsurface n as a special surface corresponding to the characteristicvalues which show the function of the optical surface with a real numbersurface number.

When the surface number of an optical surface with a real number surfacenumber is an integer n, the reliability of the best surface number givenby the optimization is improved by constructing the optical system withthe optical surface with a real number surface number so as to coincidewith the optical system with the special surface at the surface ndefined by the characteristic values of the optical surface with a realnumber surface number.

The efficiency of the optimization is improved by constructing theoptical system with optical surfaces with a real number surface numbercontinuous as for the real number surface numbers.

The efficiency of the optimization is improved by constructing theoptical system with optical surfaces with a real number surface numbersmooth as for the real number surface numbers.

When the real number surface number is between n and n+1, one method toconstruct the optical system with the optical surfaces with a realnumber surface number is to set up the optical surfaces n and n+1 asspecial surfaces corresponding to the real number surface number and thecharacteristic values of the optical surfaces with a real number surfacenumber.

When the real number surface number is between n and n+1, another methodto construct the optical system with the optical surfaces with a realnumber surface number is to insert one or more imaginary surfacesbetween the optical surfaces n and n+1 and to set up these imaginarysurfaces as special surfaces corresponding to the real number surfacenumber and the characteristic values of the optical surfaces with a realnumber surface number.

When the real number surface number is between n and n+1, by insertingtwo imaginary surfaces between the optical surfaces n and n+1, when thesurface number of an optical surface with a real number surface numberis an integer n, the construction of the optical system with the opticalsurface with a real number surface number coincides with the opticalsystem with the special surface at the surface n defined by thecharacteristic values of the optical surface with a real number surfacenumber.

The inserted two imaginary surfaces have the same base sphere and theseparation of these imaginary surfaces is 0.

The construction of the imaginary surfaces needs to have the followingfeatures. When the imaginary surfaces move between optical surfaces nand n+1, the curvature C of the base sphere changes from the curvatureC_(n) of the optical surface n to the curvature C_(n+1) of the opticalsurface n+1, the refractive index N between the imaginary surfaceschanges from the front-side index N_(n) of the optical surface n to therear-side index N′_(n+1) of the optical surface n+1, and the characterof the special surface with a real number surface number is transferredfrom the rear-side imaginary surface to the front-side imaginarysurface.

One method of the definition which fulfils the above-mentionedconditions is the following. When the distance between the opticalsurfaces n and n+1 is d, a+b=1, the distance from the optical surface nto the imaginary surfaces is a·d, and the distance from the imaginarysurfaces to the optical surface n+1 is b·d, the curvature C of the basesphere of imaginary surfaces and the refractive index N between theimaginary surfaces are determined as

C=b·C _(n) +a·C _(n+1)

N=b·N _(n) +a·N′ _(n+1)

When the refractive index between optical surfaces n and n+1 is denotedas N′_(n), and x and y are coordinates on a plane perpendicular to theoptical axis, the aspheric displacement z₁(x,y) of the front-sideimaginary surface and the aspheric displacement z₂(x,y) of the rear-sideimaginary surface are determined as

when N>N′_(n),

z ₁(x,y)=a·z(x,y)

z ₂(x,y)=−b·z(x,y)

when N<N′_(n),

z ₁(x,y)=−a·f(x,y)

z ₂(x,y)=b·f(x,y)

and the phase difference p₁(x,y) of the front-side imaginary surface andthe phase difference p₂(x,y) of the rear-side imaginary surface aredetermined as

p ₁(x,y)=a·p(x,y)

p ₂(x,y)=b·p(x,y)

The coefficient “a”, which shows a position of the imaginary surfaces,is a function of the real value surface number. One method to make theposition of the imaginary surfaces a smooth function of the real numbersurface number is to determine the coefficient “a” with a smoothfunction a(r) which satisfies

a(0)=0,a(1)=1,a′(0)=0, and a′(1)=0,

where a′(r) is the differential of a(r).

With the proposed method the surface number of a special surface can beincluded into the design process as a real value design parameter andthe best surface number can be determined by the optimization. Comparingwith the conventional procedure, in which some locations of specialsurfaces in the optical system are tried and the best result among themis chosen, the proposed system can find the best surface number within amuch shorter time.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows imaginary surfaces in an air space just after a glassspace.

FIG. 2 shows imaginary surfaces in an air space just before a glassspace.

FIG. 3 shows imaginary surfaces in a glass space just before an airspace.

FIG. 4 shows imaginary surfaces in a glass space just after an airspace.

FIG. 5 shows the explanation of the notation.

FIG. 6 shows imaginary surfaces at the center of an air space.

FIG. 7 shows Imaginary surfaces at the center of a glass space.

FIG. 8 shows the summary of the procedure.

FIG. 9 shows the starting lens for the examples.

FIG. 10 shows the relation between the surface number of the asphere andthe merit function value.

FIG. 11 shows the best solution with the asphere at the surface 1.

FIG. 12 shows the real number surface number for each solution of theglobal optimization.

FIG. 13 shows the best solution with the aspheres at the surfaces 3, 5,10.

FIG. 14 shows the best solution with the asphere with DOE at the surface1.

DETAILED DESCRIPTION OF THE INVENTION

At first the terminology is explained. The optical system consists ofrefractive or reflective surfaces, and the space between two opticalsurfaces is filled with a homogeneous or inhomogeneous material. If theoptical system is axially symmetric, the axis of symmetry is called theoptical axis. Even if the optical system is axially non-symmetric, thepath of the representative ray can be thought as the optical axis in abroad sense. Optical surfaces are classified to 2 classes. One isspheres which have their centers on the optical axis and the other isspecial surfaces. Examples of special surfaces are aspheres, diffractiveoptical elements (DOEs), and decentered surfaces. Aspheres and DOEs areeither axially symmetric or axially non-symmetric. Decentered surfacescan be thought as axially non-symmetric aspheres. An optical surface canbe both an asphere and a DOE at the same time. Optical surfaces aregiven surface numbers in the sequence of the ray path.

The surface numbers of special surfaces are integers by nature. Thismethod extends the surface number of a special surface to a real numberand defines the construction of the optical system with the opticalsurfaces with real number surface numbers. There is arbitrariness inthis definition as long as the optimization is effective for finding thebest surface numbers of special surfaces.

In the following description an optical surface with a real numbersurface number is called a floating surface. A floating surface hascharacteristic values which show the function of the surface. Oneexample of the characteristic values is the aspheric displacementz(x,y). Here x and y are coordinates on a plane perpendicular to theoptical axis. The function of a DOE is expressed with the phasedifference p(x,y).

When the surface number of a floating surface is an integer n, aprocedure is defined to set up the optical surface n as a specialsurface according to the characteristic values of the floating surface.This definition does not always mean that the same characteristic valuesare given to the optical surface n as the floating surface, but themeaning of the characteristic values of the floating surface is given bythis definition. Therefore there is arbitrariness in this definition.

The construction of the optical system with floating surfaces is definedfor any real number surface numbers. When the surface number of afloating surface is an integer n, the construction of the optical systemwith the floating surface should coincide with the optical system withthe special surface at the surface n defined by the characteristicvalues of the floating surface. If this is realized, the best surfacenumber determined by the optimization is more reliable. When theconstruction of the optical system with floating surfaces is continuousand smooth as for the real number surface numbers, the optimizationbecomes more efficient.

When the real number surface number is between n and n+1, one method todefine the optical system with the floating surface is to set up theoptical surfaces n and n+1 as special surfaces defined by the surfacenumber and the characteristic values of the floating surface.

When the real number surface number is between n and n+1, another methodto define the optical system with the floating surface is to insert oneor more imaginary surfaces between the optical surfaces n and n+1 and toset up these imaginary surfaces as special surfaces defined by thesurface number and the characteristic values of the floating surface.

Here the method to insert two imaginary surfaces will be shown. Thismethod enables that, when the surface number of a floating surface is aninteger n, the construction of the optical system with the floatingsurface coincides with the optical system with the special surface atthe surface n defined by the characteristic values of the floatingsurface.

The inserted two imaginary surfaces have the same base sphere and theseparation of these imaginary surfaces is 0. When the asphericdisplacement z(x,y) of the floating surface is identically 0, the twoimaginary surfaces coincide with the base sphere. Then the inserted twoimaginary surfaces coincide with each other and the optical system isequivalent to the case without the floating surface.

In cases where the position of the imaginary surfaces coincides with anoptical surface in the optical system, the condition for the equivalenceto the ordinary asphere, DOE, or decentered surface is considered. Thecases are classified to the cases with the imaginary surfaces just afteror just before the optical surface and the cases with the imaginarysurfaces in an air space or in a glass space. For each case the basesphere of the imaginary surfaces coincides with the optical surface.

In FIG. 1 the imaginary surfaces are in an air space just after a glassspace. The surface A is both the rear-side optical surface of the glassspace and the front-side imaginary surface. The surface B is therear-side imaginary surface and the surface C is the rear-side opticalsurface of the air space. In this case the material between theimaginary surfaces is the material of the glass space, the front-sideimaginary surface is the sphere, and the character of the floatingsurface is given to the rear-side imaginary surface.

In FIG. 2 the imaginary surfaces are in an air space just before a glassspace. The surface A is the front-side imaginary surface and the surfaceB is both the rear-side imaginary surface and the front-side opticalsurface of the glass space. The surface C is the rear-side opticalsurface of the glass space. In this case the material between theimaginary surfaces is the material of the glass space, the rear-sideimaginary surface is the sphere, and the character of the floatingsurface is given to the front-side imaginary surface.

In FIG. 3 the imaginary surfaces are in a glass space just before an airspace. The surface A is the front-side imaginary surface and the surfaceB is both the rear-side imaginary surface and the front-side opticalsurface of the air space. The surface C is the rear-side optical surfaceof the air space. In this case the material between the imaginarysurfaces is air, the rear-side imaginary surface is the sphere, and thecharacter of the floating surface is given to the front-side imaginarysurface.

In FIG. 4 the imaginary surfaces are in a glass space just after an airspace. The surface A is both the front-side optical surface of the glassspace and the front-side imaginary surface. The surface B is therear-side imaginary surface and the surface C is the rear-side opticalsurface of the glass space. In this case the material between theimaginary surfaces is air, the front-side imaginary surface is thesphere, and the character of the floating surface is given to therear-side imaginary surface.

When the position of the imaginary surfaces coincides with an opticalsurface in the optical system, the construction of the imaginarysurfaces is summarized as follows. The base sphere of the imaginarysurfaces coincides with the optical surface. The material between theimaginary surfaces coincides with the material of the opposite side ofthe imaginary surfaces over the optical surface. When the imaginarysurfaces are just after the optical surface, the character of thefloating surface is given to the rear-side imaginary surface. When theimaginary surfaces are just before the optical surface, the character ofthe floating surface is given to the front-side imaginary surface.

When the imaginary surfaces move between optical surfaces n and n+1, theconstruction of the imaginary surfaces changes as follows. In FIG. 5 thenecessary notation is explained. The curvature C of the base spherechanges from the curvature C_(n) of the optical surface n to thecurvature C_(n+1) of the optical surface n+1. The refractive index Nbetween the imaginary surfaces changes from the front-side index N_(n)of the optical surface n to the rear-side index N′_(n+1) of the opticalsurface n+1. The character of the floating surface is transferred fromthe rear-side imaginary surface to the front-side imaginary surface.

If the change of the curvature of the base surface of the imaginarysurfaces, the refractive index between the imaginary surfaces and thedistribution of the character of the floating surface over the imaginarysurfaces is a smooth function of the position of the imaginary surfaces,the optimization with the imaginary surfaces becomes efficient. Oneexample of such functions is the linear interpolation, although thepossible functions are not restricted to this example. Suppose that thedistance between the optical surfaces n and n+1 is d, the distance fromthe optical surface n to the imaginary surfaces is a·d, and the distancefrom the imaginary surfaces to the optical surface n is b·d. Here a+b=1.The curvature C of the base sphere of imaginary surfaces and therefractive index N between the imaginary surfaces are determined asfollows.

C=b·C _(n) +a·C _(n+1)

N=b·N _(n) +a·N′ _(n+1)

When the refractive index between optical surfaces n and n+1 is denotedas N′_(n), the aspheric displacement z₁(x,y) of the front-side imaginarysurface and the aspheric displacement z₂(x,y) of the rear-side imaginarysurface are determined as follows.

When N>N′_(n),

z ₁(x,y)=a·z(x,y)

z ₂(x,y)=−b·z(x,y)

When N<N′_(n),

z ₁(x,y)=−a·z(x,y)

z ₂(x,y)=b·z(x,y)

By this definition, when N>N′_(n),

z ₁(x,y)−z ₂(x,y)=z(x,y)

when N<N′_(n),

z ₁(x,y)−z ₂(x,y)=−z(x,y)

The phase difference p₁(x,y) of the front-side imaginary surface and thephase difference p₂(x,y) of the rear-side imaginary surface aredetermined as follows.

p ₁(x,y)=a·p(x,y)

p ₂(x,y)=b·p(x,y)

By this definition,

p ₁(x,y)+p ₂(x,y)=p(x,y)

FIG. 6 is an example of imaginary surfaces at the center of an airspace, and FIG. 7 is an example of imaginary surfaces at the center of aglass space.

When a=0 and b=1, the position of the floating surface coincides withthe optical surface n and the aspheric displacement becomes:

When N_(n)>N′_(n),

z ₂(x,y)=−z(x,y)

When N_(n)<N′_(n),

z ₂(x,y)=z(x,y)

When a=1 and b=0, the position of the floating surface coincides withthe optical surface n+1 and the aspheric displacement becomes:

When N′_(n+1)>N′_(n),

z ₁(x,y)=f(x,y)

When N′_(n+1)<N′_(n),

z ₁(x,y)=−f(x,y)

In this way, when the refractive index of the rear-side of the opticalsurface is smaller than the refractive index of the front-side, theaspheric displacement of the imaginary surface has the opposite sign ofthe aspheric displacement of the floating surface. The meaning of theaspheric displacement of the floating surface is given by this relation.

The coefficient “a” which shows the position of the imaginary surfacesis a function of the real number surface number. When the real numbersurface number is an integer n, the position of imaginary surfacescoincides with the optical surface n and the surface is equivalent tothe ordinary asphere, DOE, or decentered surface. In order that theconstruction of the optical system with a floating surface is continuousto the real number surface number, the coefficient “a” need to be acontinuous function of the fraction r of the real number surface numbersuch that,

a(0)=0

a(1)=1

The simplest among these functions is,

a(r)=r

But with this function the position of imaginary surfaces is not asmooth function of the real number surface number at integer numbers,because the surface separations before and after optical surfaces arenot generally equal. For keeping the efficient optimization, theaberrations of the optical system should be smooth functions of designparameters. One method to eliminate this defect is to use a smoothfunction such that,

a′(0)=0

a′(1)=0

where a′(r) is the differential of a(r). Then the position of imaginarysurfaces is always a smooth function of the real number surface number.Examples of these functions are,

a(r)=(1−cos(pi·r))/2

a(r)=2r ² −r ⁴

In FIG. 8 the procedure is summarized. The position of the imaginarysurfaces is determined from the real number surface number. Thestructure of the imaginary surfaces is determined from the position ofthe imaginary surfaces. Then the optical system with the imaginarysurfaces is evaluated. The best real number surface number is determinedas a result of the optimization by adding the real number surface numberand characteristic values of the floating surface to the independentvariables. The determined surface number is not an integer but a realnumber in general. Then the optical system needs to be optimized afterthe surface number is fixed to the nearest integer.

More than one floating surfaces can be included in an optical system. Inthis case it is necessary to avoid more than one floating surfaces tostay between optical surfaces n and n+1. This can be controlled in theoptimization with the constraints on the real number surface numbers.

EXPERIMENTAL EXAMPLE 1

FIG. 9 shows the starting lens for the following examples. For this lensFNO is 2.5, the focal length is 50 mm, and the field angle is 11.3 deg.The global optimization is used in the following examples. The purposeof the optimization is to find the minimum point of the merit function.The local optimizer finds a local minimum near the starting point. Theglobal optimizer finds many local minima in a wide region over thenearest local minimum. Then the solution that has the minimum meritfunction value is chosen from these solutions.

In example 1 the lens of FNO=2.0, the focal length=50 mm, and the fieldangle=14.0 deg is designed by using 1 asphere. For seeing the effect ofthe asphere, lenses with an asphere at each of 10 surfaces weredesigned. 20 solutions were found with the global optimization and thesolution with the minimum merit function value was chosen. FIG. 10 showsthe relation between the surface number of the asphere and the meritfunction value of the best solution. FIG. 11 shows the best solutionwith the asphere at the surface 1. The merit function of this solutionis 0.000616.

As the design with the real number surface number, the initial value 10was chosen for the real number surface number and 20 solutions werefound with the global optimization. FIG. 12 shows the real numbersurface number for each solution of the global optimization. Then thesolutions were optimized after the surface number was fixed to thenearest integers. The best solution has the asphere at the surface 4 andthe merit function value is 0.000621. Although the initial value of thereal number surface number is 10, the good solution with the asphere atthe surface 4 was found automatically.

EXPERIMENTAL EXAMPLE 2

In example 2 the lens of FNO=1.6, the focal length=50 mm, and the fieldangle=16.7 deg is designed by using 3 aspheres. In the case of 3aspheres the combination of asphere surface numbers is 120 and it is notpractical to try all the combinations. As the design with the realnumber surface numbers, the initial values 1, 5, 9 were chosen for thereal number surface numbers and the best solution has the aspheres atthe surfaces 3, 5, 10 and the merit function value is 0.000910. FIG. 13shows this solution. The merit function of the best solution withaspheres at surfaces 1, 5, 9 is 0.001261. By the design with the realnumber surface numbers, the better combination of asphere surfacenumbers was found automatically.

EXPERIMENTAL EXAMPLE 3

In example 3 the lens of FNO=2.0, the focal length=50 mm, and the fieldangle=14.0 deg is designed by using 1 asphere with DOE. The initialvalue 10 was chosen for the real number surface number and 20 solutionswere found with the global optimization. The best solution has theasphere at the surface 1 and the merit function value is 0.000476. FIG.14 shows this solution. It was shown that this method is also effectivefor the design with the aspheres with DOE.

1. A method of optimizing an optical system, comprising the steps of:extending the surface number of a special surface, which is differentfrom a sphere, to a real number, defining the construction of theoptical system with real number surface numbers of special surfaces, anddetermining the best surface numbers of special surfaces by includingthe real number surface numbers as independent variables of theoptimization of the optical system.
 2. A method of optimizing an opticalsystem according to claim 1, further comprising the feature such that:when the surface number of an optical surface with a real number surfacenumber is an integer n, a procedure is defined to set up the opticalsurface n as a special surface corresponding to the characteristicvalues which show the function of the optical surface with a real numbersurface number.
 3. A method of optimizing an optical system according toclaim 2, further comprising the feature such that: when the surfacenumber of an optical surface with a real number surface number is aninteger n, the optical system with the optical surface with a realnumber surface number is constructed so as to coincide with the opticalsystem with the special surface at the surface n defined by thecharacteristic values of the optical surface with a real number surfacenumber.
 4. A method of optimizing an optical system according to claim2, further comprising the feature such that: the optical system withoptical surfaces with a real number surface number is constructedcontinuous as for the real number surface numbers.
 5. A method ofoptimizing an optical system according to claim 2, further comprisingthe feature such that: the optical system with optical surfaces with areal number surface number is constructed smooth as for the real numbersurface numbers.
 6. A method of optimizing an optical system accordingto claim 2, further comprising the feature such that: when the realnumber surface number is between n and n+1, the optical system with theoptical surfaces with a real number surface number is constructed bysetting up the optical surfaces n and n+1 as special surfacescorresponding to the real number surface number and the characteristicvalues of the optical surfaces with a real number surface number.
 7. Amethod of optimizing an optical system according to claim 2, furthercomprising the feature such that: when the real number surface number isbetween n and n+1, the optical system with the optical surfaces with areal number surface number is constructed by inserting one or moreimaginary surfaces between the optical surfaces n and n+1 and by settingup these imaginary surfaces as special surfaces corresponding to thereal number surface number and the characteristic values of the opticalsurfaces with a real number surface number.
 8. A method of optimizing anoptical system according to claim 7, further comprising the feature suchthat: when the real number surface number is between n and n+1, theoptical system with the optical surfaces with a real number surfacenumber is constructed by inserting two imaginary surfaces between theoptical surfaces n and n+1.
 9. A method of optimizing an optical systemaccording to claim 8, further comprising the feature such that: theinserted two imaginary surfaces have the same base sphere and theseparation of these imaginary surfaces is
 0. 10. A method of optimizingan optical system according to claim 9, further comprising the featuresuch that: when the imaginary surfaces move between optical surfaces nand n+1, the curvature C of the base sphere changes from the curvatureC_(n) of the optical surface n to the curvature C_(n+1) of the opticalsurface n+1, the refractive index N between the imaginary surfaceschanges from the front-side index N_(n) of the optical surface n to therear-side index N′_(n+1) of the optical surface n+1, and the characterof the special surface with a real number surface number is transferredfrom the rear-side imaginary surface to the front-side imaginarysurface.
 11. A method of optimizing an optical system according to claim10, further comprising the feature such that: when the distance betweenthe optical surfaces n and n+1 is d, a+b=1, the distance from theoptical surface n to the imaginary surfaces is a·d, and the distancefrom the imaginary surfaces to the optical surface n+1 is b d, thecurvature C of the base sphere of imaginary surfaces and the refractiveindex N between the imaginary surfaces are determined asC=b·C _(n) +a·C _(n+1)N=b·N _(n) +a·N′ _(n+1) when the refractive index between opticalsurfaces n and n+1 is denoted as N′_(n), and x and y are coordinates ona plane perpendicular to the optical axis, the aspheric displacementz₁(x,y) of the front-side imaginary surface and the asphericdisplacement z₂(x,y) of the rear-side imaginary surface are determinedas when N>N′_(n),z ₁(x,y)=a·z(x,y)z ₂(x,y)=−b·z(x,y) when N<N′_(n),z ₁(x,y)=−a·f(x,y)z ₂(x,y)=b·f(x,y) and the phase difference p₁(x,y) of the front-sideimaginary surface and the phase difference p₂(x,y) of the rear-sideimaginary surface are determined asp ₁(x,y)=a·p(x,y)p ₂(x,y)=b·p(x,y)
 12. A method of optimizing an optical system accordingto claim 9, further comprising the feature such that: when the distancebetween the optical surfaces n and n+1 is d and the fraction of the realnumber surface number is r, the distance from the optical surface n tothe imaginary surfaces is determined as a(r)·d with a smooth functiona(r) which satisfiesa(0)=0,a(1)=1,a′(0)=0, and a′(1)=0, where a′(r) is the differential ofa(r).
 13. A method of optimizing an optical system having at least firstand second optical surfaces, comprising: inserting a special surface atan arbitrary position between the first and second optical surfaces, thespecial surface having at least one special characteristic and at leastone characteristic value; associating a real number surface number withthe special surface, the real number surface number relating to theposition between the first and second optical surfaces; forming a firstplurality of independent variables, the first plurality including atleast the real number surface number and the at least one characteristicvalue; forming a merit function from at least the first plurality ofindependent variables; and optimizing the merit function to form a firstoptimized merit function and a first plurality of optimized independentvariables.
 14. The method of claim 13, wherein: the at least one specialcharacteristic is an asphere; and the at least one characteristic valueincludes at least one aspheric coefficient.
 15. The method of claim 13,wherein: the at least one special characteristic is an diffractiveoptical element; and the at least one characteristic value includes atleast one phase difference.
 16. The method of claim 13, wherein: the atleast one special characteristic is a decentered surface; and the atleast one characteristic value includes at least one displacement. 17.The method of claim 13, further comprising: determining an optimizedspecial surface having an optimized real number surface number and anoptimized at least one characteristic value from the first plurality ofoptimized independent variables; determining an optimized position ofthe optimized special surface between the first and second opticalsurfaces from the optimized real surface number; determining a nearsurface to be the closer of the first and second optical surfaces to theoptimized special surface; assigning the special characteristic and atleast one unoptimized characteristic value to the near surface; forminga second plurality of independent variables, the second pluralityincluding the at least one unoptimized characteristic value; forming asecond merit function from at least the second plurality of independentvariables; and optimizing the second merit function to form a secondoptimized merit function and a second plurality of optimized independentvariables.
 18. The method of claim 13, wherein: the special surfacecomprises a first and a second imaginary surface both disposed at thespecial surface with a separation of 0; the first and second imaginarysurfaces have first and second curvatures, respectively, which are equalto each other; and the first and second imaginary surfaces both have theat least one special characteristic of the special surface.
 19. Themethod of claim 18, wherein: the first plurality of independentvariables further includes a special surface position; during theoptimizing step, the special surface position is optimized and describesat least one intermediate location between the first and second opticalsurfaces; and as the at least one intermediate location varies from thefirst to the second optical surface. the curvatures of both the firstand second imaginary surfaces vary from a curvature of the first opticalsurface to a curvature of the second optical surface; and a refractiveindex between the first and second imaginary surfaces varies from arefractive index adjacent to the first optical surface and facing awayfrom the second optical surface to a refractive index adjacent to thesecond optical surface and facing away from the first optical surface.20. The method of claim 19, wherein the special surface position is asmooth function of the real number surface number.
 21. The method ofclaim 20, wherein the real number surface number equals an integer whenthe special surface position coincides with the first or second opticalsurfaces.
 22. The method of claim 21, wherein the first derivative ofthe special surface position with respect to the real number surfacenumber equals 0 when the special surface position coincides with thefirst or second optical surfaces.
 23. The method of claim 20, wherein:the real number surface number equals a first integer when the specialsurface position coincides with the first optical surface; the realnumber surface number equals a second integer when the special surfaceposition coincides with the second optical surface; and the secondinteger equals the first integer plus
 1. 24. A method of optimizing anoptical system having at least a first optical surface and a secondoptical surface, comprising: inserting a floating surface between thefirst and the second optical surfaces, the floating surface comprising afirst imaginary surface and a second imaginary surface separated fromthe first imaginary surface by zero; forming a plurality of independentvariables, the plurality including at least a position of the floatingsurface; forming a merit function that includes at least the pluralityof independent variables; and optimizing the merit function, so thatduring the optimization the floating surface has a location described byat least one intermediate location between the first and second opticalsurfaces; wherein: a quantity a is defined as varying from 0 to 1 anddescribing the at least one intermediate location, with (a=0) describingthe first optical surface and (a=1) describing the second opticalsurface; a quantity C_(n) is defined as a curvature of the first opticalsurface; a quantity C_(n+1) is defined as a curvature of the secondoptical surface; a curvature C of both the first and second imaginarysurfaces is given by a quantity ((1−a)·C_(n))+(a·C_(n+1)); a quantityN_(n) is defined as a refractive index adjacent to the first opticalsurface and facing away from the second optical surface; a quantityN′_(n+1) is defined as a refractive index adjacent to the second opticalsurface and facing away from the first optical surface; a refractiveindex N between the first and second imaginary surfaces is given by aquantity ((1−a)·N_(n))+(a·N′_(n+1)); a quantity N′_(n) is defined as arefractive index between the first and second optical surfaces; aquantity z(x,y) is defined as an aspheric displacement of the floatingsurface; an aspheric displacement z₁(x,y) of the first imaginary surfaceis given by a quantity (−a·z(x,y)) if (N<N′_(n)) or by a quantity(a·z(x,y)) if (N>N′_(n)); an aspheric displacement z₂(x,y) of the secondimaginary surface is given by a quantity ((1−a)·z(x,y)) if (N<N′_(n)) orby a quantity (−(1−a)·z(x,y)) if (N>N′_(n)); a quantity p(x,y) isdefined as a phase difference of the floating surface; a phasedifference p₁(x,y) of the first imaginary surface is given by a quantity(a·p(x,y)); and a phase difference p₂(x,y) of the second imaginarysurface is given by a quantity ((1−a)·p(x,y)).
 25. The method of claim24, wherein the quantity a varies linearly with the separation betweenthe first optical surface and the first imaginary surface.